Inverse problem of the Holling-Tanner model and its solution

In this paper we undertake to consider the inverse problem of parameter identification of nonlinear system of ordinary differential equations for a specific case of complete information about solution of the Holling-Tanner model for finite number of points for the finite time interval. In this model the equations are nonlinearly dependent on the unknown parameters. By means of the proposed transformation the obtained equations become linearly dependent on new parameters functionally dependent on the original ones. This simplification is achieved by the fact that the new set of parameters becomes dependent and the corresponding constraint between the parameters is nonlinear. If the conventional approach based on introduction of the Lagrange multiplier is used this circumstance will result in a nonlinear system of equations. A novel algorithm of the problem solution is proposed in which only one nonlinear equation instead of the system of six nonlinear equations has to be solved. Differentiation and integration methods of the problem solution are implemented and it is shown that the integration method produces more accurate results and uses less number of points on the given time interval. Keywords-Parameter estimation, Goal function, Absolute error curves, Inverse method, HollingTanner model, Least square method, Differentiation method, Integration method


I. INTRODUCTION
The numerical evaluation of known coefficient of a dynamical system i.e. the problem of dynamical system identification, is one of the most important problem of the mathematical biology [1], ecology [2], [3], [4], etc. Usually, to identify a dynamics of a system, it is necessary to have certain statistical information for time values about the unknown functions of this system.In the present paper we consider the inverse problem of parameter identification of the Holling-Tanner predator-prey model [5], [6].This model is widely used in mathematical biology, for example, in the study of transmissible disease [7].Several investigations have been done by various researchers on the mite-spider-mite, lynx-hare and sparrow-hawksparrow competition [8], [9], [10].In [11], the authors proposed a method consisting in the direct integration of a given dynamical system with the subsequent application of quadrature rules and the least square method [12], [13] provided that there is complete statistical information about the unknown function.In this paper, we assume that the complete information about the competing species is available and the two methods of solution, differentiation and integration methods, are proposed.The problem of the Holling-Tanner model identification has its specifics, because it nonlinearly depends on the unknown parameters.It is possible to transform this model to a new form where the equations of the system linearly depends on the set of new parameters.These new parameters are not independent and we need to consider the constraint between the parameters, which are nonlinear.The Holling-Tanner model has only one constraint and hence, can be simply treated by a novel method developed by the authors.The theoretical considerations are accompanied by numerical examples where the developed algorithm is tested for both differentiation and integration methods of solution.It is shown that the integration methods is more accurate than the differentiation one and needs less amount of experimental information.
• Initial value problem (1) has the positive steady-state solution [15] (x, ỹ) which cor-responds to either stable focus or stable node critical point depending on b 1 , • • • b 6 so that: where • Initial value problem (1) has unstable steadystate solution which corresponds to the saddle critical point.

PROBLEM
Assume that solution of initial problem (1), x(t) and y(t) is given on the finite time interval t ∈ [0, T ] with initial t = 0 and terminal t = T time instants in N + 1 equispaced time instants t i = T N i ∈ [0, T ]: +C 4 (x(t)) + C 5 (x 2 (t)) + (−x(t) ẋ(t)) = 0, (4) Biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 where Considering x(t) and y(t) in time instants t = t i we obtain the following overdetermined system of N + 1 linear algebraic equations: where i , and are (N + 1) × 1-vector columns.Hence, the unknown parameters C 1 , C 2 , C 3 , C 4 and C 5 can be found by, for example, the least squares method [19] by means of the constrained minimization of function G 1 : This problem can be solved providing that vectors f 1 , • • • , f 5 are linearly independent in (6).The last term contains the Lagrange multiplier λ and the constraint between coefficients Moreover, the second equation of system (1) can be rewritten in time instants t = t i as the following overdetermined system of N + 1 linear algebraic equations: where That is why coefficients C 6 , C 7 can be found by application of the least square method by means This problem can be solved providing that vectors f 7 and f 8 are linearly independent of (8).
Remark 2. In vectors f 3 , f 6 the component ẋi , and in vector f 9 the components ẏi are calculated by means of numerical differentiation of x i , y i with respect to time t and that is why the proposed method is called the differential method of identification.1) can be identified by the least square method [19] if Proof: Integrating expression (4) with respect to time t ∈ [0, T ] we obtain Integrating second equation of system 5 with respect to time t ∈ [0, T ] we have Biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057Performing all integrations in ( 9) and ( 10) from 0 to t j ∈ [0, T ] we obtain the following overdetermined systems of N + 1 linear algebraic equations where are the (N + 1) × 1-vector columns.Now applying the method used in Theorem 1 we prove the Corollary.
Remark 4. In vector g 1 , g 2 , g 4 , g 5 , g 7 , g 8 the integrals are calculated by means of numerical integration of x i , y i and their combinations with respect to time t and that is why the proposed method is called the integration method of identification.
Remark 5. Note that expressions (5), ( 7) and ( 11) are linear with respect to unknown constants C 1 , • • • , C 7 .Direct use of the constraint minimization using the Lagrange multiplier with constraint: produces nonlinear system of equations for determination of six unknowns Thus the search is performed in six-dimensional space of parameters and hence this method substantially complexifies the solution procedure.Determination of parameters and C 6 and C 7 needs solution of linear system of two algebraic equations.In the next section we describe an original problem solution algorithm reducing the search space dimension to one and using only linear matrix manipulations in the process of solution, which substantially simplifies and accelerates the problem solution.

IV. SOLUTION OF THE PARAMETER IDENTIFICATION PROBLEM
There are four original independent parameters (b 1 , b 2 , b 3 , b 4 ) in the first equation of (1).First four Cparameters (C 1 , C 2 , C 3 , C 4 ) depend on bparameters so that there is one-to-one correspondence between them.The parameter C 5 depends on the first four C-parameter as follows: Hence, it is possible to consider (C 1 , C 2 , C 3 , C 4 ) as independent parameters and introduce new name for the dependent parameter C 5 = −λ.The novel algorithm will be considered in detail for the differentiation method of solution, i.e. with f 1,••• ,9 -vector columns(see expression ( 5) and (7).The integration method of solution uses the same algorithm in which f 1,••• ,9 -vector columns are changed to g 1,••• ,9 -ones (see (11)).Parameter λ will be selected from the given interval λ ∈ [λ min , λ max ] and substituted in goal function G 3 which is composed as follows and subjected to minimization.In expression (14), parameter λ is considered as constant at every minimization and minimization itself is performed with respect to parameters C 1 , C 2 , C 3 , C 4 .Solution of this problem is given by the following formula where In expression (15) it is possible to calculate 1 × (N +1)vector row (L T 1 L 1 ) −1 L T 1 only once and Biomath 7 (2018), 1812057, http://dx.doi.org/10.11145/j.biomath.2018.12.057 after that perform its multiplication by (N +1)×1vector row R(λ), which is very fast operation.Components of vector C(λ) and C 5 = −λ are substituted in the constraint (12) to obtain the following nonlinear scalar equation which is solved with respect to λ.All roots of Equation ( 17) are found (sometimes to find all the roots it is necessary to expand the interval λ ∈ [λ min , λ max ] to the left or to the right or to both sides).After finding a particular root λ the corresponding b-parameters are calculated as follows: (See ( 4 8. Solution of this problem is given by the formulas: where Comparison of original graphs with graphs obtained by numerical solution of initial problem (1) with the same initial conditions but with estimated parameters is shown in Figure 5 and Figure 6.As we see the estimated parameters give very good estimation of the process dynamics.The

y 2
are linearly dependent.Parameters b 5 and b 6 of the abovementioned model can be identified by the abovementioned method if (N + 1) × 1-vector columns ti 0 y(τ )dτ and ti 0 (τ ) x(τ ) dτ are linearly dependent.
)).The estimations of b-parameters are obtained from the proper selection of root λ = λ: b1 = b 1 ( λ), b2 = b 2 ( λ), b3 = b 3 ( λ), b4 = b 4 ( λ) (19) (one of the criteria of the correct choice of λ must be positiveness of all estimated b parameters, see Numerical Examples).Parameters b 5 and b 6 are estimated by means of minimization of the goal function of Equation

Fig. 5 .
Fig. 5. Graph of original solution x = x(t) (dots) and solution with estimated parameters (solid line)